# Complex Links and Algebraic Multiplicities

Abstract:

In this talk I will consider two invariants associated to pairs of strata $X < Y$ in an analytic stratification of some ambient complex projective variety $W$. The main result states that if the closures $\overline{X} \subset \overline{Y}$ are subvarieties of $W$ with $\overline{X}$ irreducible, then the Hilbert-Samuel (or algebraic) multiplicity of $\overline{Y}$ along $\overline{X}$ equals the Euler characteristic of the space obtained by intersecting $Y$ with the complex link of $X$ in $W$. I will also discuss the ingredients used to prove this result, and in particular will present a local Lefschetz hyperplane theorem for such complex linking spaces. Applications of this result include a new algebraic formula for MacPherson's local Euler obstruction of $W$. This talk is based on recent joint work with Vidit Nanda [Oxford] which is available on the arxiv (https://arxiv.org/abs/2006.10452).

Venue:

Seminar Room 1.33, Hanna Neumann Building 145 - for up to 25 people.

Participation via Zoom will also be avalible.